{"id":234237,"date":"2025-06-14T02:45:41","date_gmt":"2025-06-14T02:45:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234237"},"modified":"2025-06-14T02:45:43","modified_gmt":"2025-06-14T02:45:43","slug":"find-the-exact-value-of-cos-75-degrees-in-terms-of-radicals","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/find-the-exact-value-of-cos-75-degrees-in-terms-of-radicals\/","title":{"rendered":"Find the exact value of cos 75 degrees in terms of radicals"},"content":{"rendered":"\n

Find the exact value of cos 75 degrees in terms of radicals. * (
)\/4 (
)\/4 (
)\/4 None of the above<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer:<\/strong> The correct option is (\u221a6 – \u221a2)\/4<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To find the exact value of cos(75\u00b0) in terms of radicals, we can express 75\u00b0 as a sum of two special angles for which the exact trigonometric values are well-known. The most convenient way to do this is to write 75\u00b0 as the sum of 45\u00b0 and 30\u00b0.<\/p>\n\n\n\n

1. Decompose the Angle:<\/strong>
We can write 75\u00b0 as:
75\u00b0 = 45\u00b0 + 30\u00b0<\/p>\n\n\n\n

2. Apply the Cosine Angle Addition Formula:<\/strong>
The angle addition formula for cosine is:
cos(A + B) = cos(A)cos(B) – sin(A)sin(B)<\/p>\n\n\n\n

By substituting A = 45\u00b0 and B = 30\u00b0 into this formula, we get:
cos(75\u00b0) = cos(45\u00b0 + 30\u00b0) = cos(45\u00b0)cos(30\u00b0) – sin(45\u00b0)sin(30\u00b0)<\/p>\n\n\n\n

3. Substitute the Known Values of Special Angles:<\/strong>
We know the exact values for the sine and cosine of 45\u00b0 and 30\u00b0:<\/p>\n\n\n\n

    \n
  • cos(45\u00b0) = \u221a2 \/ 2<\/li>\n\n\n\n
  • sin(45\u00b0) = \u221a2 \/ 2<\/li>\n\n\n\n
  • cos(30\u00b0) = \u221a3 \/ 2<\/li>\n\n\n\n
  • sin(30\u00b0) = 1 \/ 2<\/li>\n<\/ul>\n\n\n\n

    Now, we substitute these values into our equation:
    cos(75\u00b0) = (\u221a2 \/ 2) * (\u221a3 \/ 2) – (\u221a2 \/ 2) * (1 \/ 2)<\/p>\n\n\n\n

    4. Simplify the Expression:<\/strong>
    First, multiply the terms:
    cos(75\u00b0) = (\u221a2 * \u221a3) \/ (2 * 2) – (\u221a2 * 1) \/ (2 * 2)
    cos(75\u00b0) = \u221a6 \/ 4 – \u221a2 \/ 4<\/p>\n\n\n\n

    Next, combine the terms over the common denominator:
    cos(75\u00b0) = (\u221a6 – \u221a2) \/ 4<\/strong><\/p>\n\n\n\n

    This result matches the third option provided in the question. Since 75\u00b0 is in the first quadrant, its cosine value must be positive. As \u221a6 \u2248 2.449 and \u221a2 \u2248 1.414, the numerator (\u221a6 – \u221a2) is positive, which is consistent with our expectation.thumb_upthumb_down<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Find the exact value of cos 75 degrees in terms of radicals. * ()\/4 ()\/4 ()\/4 None of the above The Correct Answer and Explanation is: Correct Answer: The correct option is (\u221a6 – \u221a2)\/4. Explanation: To find the exact value of cos(75\u00b0) in terms of radicals, we can express 75\u00b0 as a sum of two special […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234237","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234237","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234237"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234237\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}